Question: Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = 5,$ $\|\mathbf{c}\| = 3,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{c}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b},$ then find $\sin \theta.$
Answer: Note that vectors $\mathbf{a}$ and $\mathbf{a} \times \mathbf{b}$ are orthogonal, so
\[\|\mathbf{c}\| = \|\mathbf{a} \times (\mathbf{a} \times \mathbf{b})\| = \|\mathbf{a}\| \|\mathbf{a} \times \mathbf{b}\|.\]Also, $\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin \theta,$ so
\[3 = 1 \cdot 1 \cdot 5 \sin \theta.\]Hence, $\sin \theta = \boxed{\frac{3}{5}}.$